Saddlepoint Problems in Contifuous Time Rational Expectations Models: A General Method and Some Macroeconomic Ehamples

Abstract

The paper presents general solution methods for rational expectations models that can be represented by systems of deterministic first order linear differential equations with constant coefficients. One method is the continuous time adaptation of the method of Blanchard and Kahn. To obtain a unique solution there must be as many linearly independent boundary conditions as there are linearly independent state variables. Three slightly different versions of a well-known small open economy macroeconomic model are used to illustrate three fairly general ways of specifying the required boundary conditions. The first represents the standard case in which the number of stable characteristic roots equals the number of predetermined variables. The second represents the case when the number of stable roots exceeds the number of predetermined variables but a sufficient number of linear restrictions on the state variables at an initial date is given to guarantee a unique solution. The third represents the case when the missing initial conditions have been replaced by boundary conditions that involve linear restrictions on the values of the state variables across an initial and a future date. The method of this paper permits the numerical solution of models with large numbers of state variables. Any combination of anticipated or unanticipated, current or future and permanent or transitory shocks can be analyzed. THIS PAPER STUDIES the solution of a class of rational expectations models that can be represented by systems of deterministic first order linear differential equations with constant coefficients. This class includes virtually all deterministic continuous time rational expectations models in the macroeconomic and open economy macroeconomic literature such as Sargent and Wallace [22], Dornbusch [13], Wilson [25], Krugman [16], Dornbusch and Fischer [14], Buiter and Miller [7, 8, 9], Begg [2], and Obstfeld [20]. The two solution methods proposed in the paper handle systems with state vectors of any dimension, n. As long as the forcing variables or exogenous variables do not explode too fast, any combination of anticipated or unanticipated, current or future, and permanent or transitory shocks can be analyzed. Wilson's [25] analysis of anticipated future shocks in systems where n = 2 and Dixit's [12] method for handling unanticipated current permanent shocks are special cases of the general methods developed in this paper. When the number of predetermined or backward-looking variables (n,) equals the number of stable roots of the characteristic equation of the homogenous system and the number of non-predetermined, forward-looking or jump variables (n - n ,) equals the number of unstable roots, there is a natural way of